Harold McCord Jr. was airlifted to a Seattle hospital, where he died, authorities said.
The unidentified officer, also airlifted to Harborview Medical Center, reportedly was hit in the hand and arm. His injuries were not considered life-threatening.
The shootout took place at an apartment complex in Monroe, about 60 miles northeast of Tacoma. Police went there after determining that one of McCord's criminal associates was linked to the address, Detective Ed Troyer said.
McCord, 36, had been sentenced earlier this month to life in prison under Washington's "three strikes" law. He had previously been convicted of robbery, harassment, assault and kidnapping and recently received the life sentence, sheriff's spokeswoman Lauren Pawlawski said.
McCord's handcuffs and shackles were removed while he was in court Monday for a hearing in another case.
He ran from the courtroom and was tackled by a guard. As they were fighting, he pulled out what appeared to be a handgun. Other officers lowered their weapons and he fled, Pawlawski said.
McCord tried to hijack one vehicle, but that driver drove away. The next driver got out and the inmate drove away in his pickup truck, which was found about three miles away.
The fake gun was found in the road, officials said. It was made of cardboard from the backing of legal pads, stuffed with toilet paper and covered with newspaper blackened with ink from a ballpoint pen.